Omitting the <code>relerr</code> argument, or passing <code>relerr=0</code> (or any <code>relerr</code> less than machine precision &epsilon;&asymp;10<sup>&minus;16</sup>), corresponds to requesting [[w:Machine epsilon|machine precision]], and in practice a relative error &lt;&nbsp;10<sup>&minus;13</sup> is usually achieved. Specifying a larger value of <code>relerr</code> may sometimes improve performance (at the expense of accuracy).

+

Omitting the <code>relerr</code> argument, or passing <code>relerr=0</code> (or any <code>relerr</code> less than machine precision &epsilon;&asymp;10<sup>&minus;16</sup>), corresponds to requesting [[w:Machine epsilon|machine precision]], and in practice a relative error &lt;&nbsp;10<sup>&minus;13</sup> is usually achieved. Specifying a larger value of <code>relerr</code> may improve performance (at the expense of accuracy).

You should also compile <code>Faddeeva_w.cc</code> and link it with your program, of course.

You should also compile <code>Faddeeva_w.cc</code> and link it with your program, of course.

Omitting the relerr argument, or passing relerr=0 (or any relerr less than machine precision ε≈10−16), corresponds to requesting machine precision, and in practice a relative error < 10−13 is usually achieved. Specifying a larger value of relerr may improve performance (at the expense of accuracy).

You should also compile Faddeeva_w.cc and link it with your program, of course.

Wrappers: Matlab, GNU Octave, and Python

Wrappers are available for this function in other languages.

Matlab: A function Faddeeva_w(z, relerr), where the arguments have the same meaning as above (the relerr argument is optional) can be downloaded from Faddeeva_w_mex.cc (along with the help file Faddeeva_w.m. Compile it into an octave plugin with:

mex -output Faddeeva_w -O Faddeeva_w_mex.cc Faddeeva_w.cc

GNU Octave: A function Faddeeva_w(z, relerr), where the arguments have the same meaning as above (the relerr argument is optional) can be downloaded from Faddeeva_w_oct.cc. Compile it into a MEX file with:

(I initially used this algorithm for all z, but the continued-fraction expansion turned out to be faster for larger |z|. On the other hand, it is competitive for smaller |z|, and is significantly more accurate than the Poppe & Wijers code in some regions, e.g. in the vicinity of |z|=1. You can switch back to using Algorithm 916 for all z by changing USE_CONTINUED_FRACTION to 0 in the code.)

Note that this is SGJ's independent re-implementation of these algorithms, based on the descriptions in the papers only. In particular, we did not refer to the authors' Fortran or Matlab implementations (respectively), which are under restrictive "semifree" ACM copyright terms and are therefore unusable in free/open-source software.

Algorithm 916 requires an external complementary error function erfc(x) function for real arguments x to be supplied as a subroutine. More precisely, it requires the scaled function erfcx(x) = ex2erfc(x). Here, we include an erfcx function derived from the DERFC routine in SLATEC (modified by SGJ to compute erfcx instead of erfc), originally written by W. Fullerton at Los Alamos National Laboratory.

Test program

To test the code, a small test program is included at the end of Faddeeva_w.cc which tests w(z) against several known results (from Wolfram Alpha) and prints the relative errors obtained. To compile the test program, #define FADDEEVA_W_TEST in the file (or compile with -DFADDEEVA_W_TEST on Unix) and compile Faddeeva_w.cc. The resulting program prints SUCCESS at the end of its output if the errors were acceptable.

License

The software is distributed under the "MIT License", a simple permissive free/open-source license:

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.